Microlocal analysis of a seismic linearized inverse problem

This paper addresses the cone transform that integrates a function over the surfaces of circular cones. We derive an inversion formula for the cone transform in n-dimensional Euclidean space for cones having a fixed central axis direction, but having all opening angles and vertices. We also show that the cone transform and its dual are Fourier integral operators, and the normal operator is a pseudo-differential operator. We finally present a differential equation that the cone transform satisfies, which can be seen as a first step in describing the range of the cone transform.

The Born approximation of a potential in the context of the Calderón inverse problem is an object that can be formally defined in terms of spectral data of the Dirichlet-to-Neumann map of the corresponding Schrödinger operator. In this article, we prove, in the case of radial potentials in the Euclidean ball and any fixed energy, that the Born approximation is well-defined as a compactly supported radial distribution, and that the Calderón problem can be reformulated as recovering a potential from its Born approximation. In addition, we show that the Born approximation depends locally on the potential and captures exactly its singularities, and that the functional that maps the Born approximation to the potential is Hölder continuous. We also prove that the Born approximation converges to the potential in the high-energy limit. Moreover, we give an explicit formula for the Fourier transform of the Born approximation at any fixed energy, and illustrate how it can be used as the basis of an accurate procedure to approximate a potential from its Dirichlet-to-Neumann map.

The solution of direct and inverse transient heat conduction problems with layered materials using exponential basis functions

The linear sampling method is the oldest and most developed of the qualitative methods in inverse scattering theory. It is based on solving a linear integral equation and then using the equation s solution as an indicator function for the determination of the support of the scattering object. This book describes the linear sampling method for a variety of electromagnetic scattering problems. It presents uniqueness theorems and the derivation of various inequalities on the material properties of the scattering object from a knowledge of the far field pattern of the scattered wave. Also covered are the approximation properties of Herglotz wave functions; the behavior of solutions to the interior transmission problem, a novel interior boundary value problem; and numerical examples of the inversion scheme. Audience: This book is intended for mathematicians and engineers performing research in inverse electromagnetic scattering theory. It is also appropriate for an advanced graduate course on inverse problems. Contents: Preface; Chapter 1: Inverse Scattering in Two Dimensions; Chapter 2: Maxwell s Equations; Chapter 3: The Inverse Problem for Obstacles; Chapter 4: The Inverse Scattering Problem for Anisotropic Media; Chapter 5: The Inverse Scattering Problem for Thin Objects; Chapter 6: The Inverse Scattering Problem for Buried Objects; Bibliography; Index.

This paper presents a frequency-domain ultrasonic technique for a simultaneous determination of the thickness (h) and wave speed (c) of the individual layers comprising a multilayered medium. The layers may be ‘‘thin’’; by thin we mean that the successive reflections of an ultrasonic pulse from the two faces of a layer are nonseparable in the time domain. Plane longitudinal waves which are normally incident upon the medium are considered. A systematic analysis of the sensitivity of the complex-valued transfer function to the acoustical parameters of each layer has been carried out. An inverse algorithm, which utilizes either the Newton–Raphson or the Simplex method in conjunction with the incremental search method, has been developed to reconstruct simultaneously the thickness and phase velocity of each layer by minimizing the difference between the theoretical and the experimental results in the mean-sum-square sense; the entire complex spectrum, i.e., the amplitude as well as the phase spectrum, was used. The technique is fully automated and computer controlled and can be readily used for in situ NDE applications. Results are presented for several three-layer specimens; aluminum/water/aluminum, aluminum/water/titanium, and titanium/water/titanium. Successful inversion was obtained for the following cases (1) simultaneous determination of h and c of any one of the three layers, given h and c of the remaining two layers; (2) simultaneous measurement of the three thicknesses, given the three wave speeds; (3) simultaneous measurement of the three wave speeds, given the three thicknesses; (4) simultaneous determination of all three thicknesses and one wave speed, given the remaining two wave speeds. The precision of our measurements was found to be excellent; typically, ±3 μm in h (for h of the order of 1 mm) and ± one part per thousand in c. The accuracy was found to be about one order of magnitude lower than the precision; typically, ±10 μm in h and ±2% in c.

This reference is for an abstract only. A full paper was not submitted for this conference. Abstract AVO (amplitude-varying-with-offset) is a seismic prestack inversion technique for estimating elastic parameters of the subsurface. The seismic AVO inversion problem can be formulated as a linearized or nonlinear inverse problem. It is a multidimensional and highly ill-posed inverse problem. It is affected by strong noise and measurement uncertainty. Therefore, in seismic AVO analysis, the goal is not only to find the model that best-fits the data, but also to characterize the uncertainty of the analysis. Uncertainty characterization of seismic analysis makes the geophysical interpretation more reliable. The Bayesian formalism generally constitutes a powerful approach for solving many seismic inverse problems. It allows combining available prior knowledge with the information contained in the seismic data. The solution of a Bayesian inverse problem is given by the joint posterior distribution of model parameters. Stochastic approximations of the true posterior are exemplified by the MCMC method and more recently by the particle filtering (PF) method. For most seismic inverse problems, stochastic approximation techniques are computationally unappealing. The maximum a posteriori (MAP) is the simplest Bayesian method for estimating model parameters. It provides the explanation that maximizes the posterior distribution. The maximum likelihood (ML) is a popular non-Bayesian method for estimating model parameters. It provides the explanation that maximizes the likelihood of the data. It assumes that the model parameters are deterministic and omits the pertinent prior knowledge on them. In practice, both MAP and ML point estimators have severe problems with over-fitting the data and model order estimation. In this paper, the recently-developed variational Bayesian learning is proposed as a new method for solving the linearized seismic AVO inversion problem. Variational Bayesian learning allows the true joint posterior distribution of model parameters to be approximated by a simpler approximating ensemble for which the required inferences are tractable. The main advantage in resorting to variational Bayesian learning is its robustness to the over-fitting problem, which is major in practice. This probabilistic machine learning method allows finding an optimum balance between the representation power and the model complexity of the data.

Inverse radiation problem in one-dimensional semitransparent plane-parallel media with opaque and specularly reflecting boundaries

This paper is concerned with developing a technique for an ultrasonic determination of the acoustical properties (thickness, wave speed, density, attenuation) of the individual layer in a layered medium. The layered medium is surrounded by an identical host medium (water) of infinite extent on either side. Normally incident plane longitudinal waves are considered. Transfer functions have been calculated in time domain as well as in frequency domain. A satisfactory comparison between the theory and the experiment was observed for a three-layered medium (forward problem). An inverse problem is defined as a simultaneous determination of several acoustic parameters from the measured transfer functions. A systematic analysis of the sensitivity of the transfer functions to the acoustic parameters has been carried out. The following inverse problems have been solved for a three-layered medium: three thicknesses for a three-layered medium; and wave speed and thickness of any one of the layers (given those of the remaining two layers). Finally, relative merits of the time-domain and the frequency-domain approaches have been studied. [Work supported by Texas Advanced Technology Program.]

We study the basic properties of the $d$-plane transform on the Euclidean space as a Fourier integral operator and its application to the microlocal analysis of streaking artifacts in its filtered back-projection. The $d$-plane transform is defined by integrals of functions on the $n$-dimensional Euclidean space over all the $d$-dimensional planes, where $0<d<n$. This maps functions on the Euclidean space to those on the affine Grassmannian $G(d,n)$. This is said to be X-ray transform if $d=1$ and Radon transform if $d=n-1$. When $n=2$ the X-ray transform is thought to be measurements of computed tomography (CT) scanners. In this paper we obtain a concrete expression of the canonical relation between the $d$-plane transform and the quantitative properties of the filtered back-projection of the product of the images of the $d$-plane transform. The FBP of the product is related to the metal streaking artifacts of CT images. Our result is a generalization of recent results of Park, Choi, and Seo [Comm. Pure Appl. Math., 70 (2017), pp. 2191--2217] and Palacios, Uhlmann, and Wang [SIAM J. Math. Anal., 50 (2018), pp. 4914--4936] for the X-ray transform on the plane.

Nuclear resonant inelastic X-ray scattering (NRIXS) experiments have been applied to Earth materials, and the Debye speed is often related to the material’s seismic wave speeds. However, for anisotropic samples, the Debye speed extracted from NRIXS measurements is not equal to the Debye speed obtained using the material’s isotropic seismic wave speeds. The latter provides an upper bound for the Debye speed of the material. Consequently, the acoustic wave speeds estimated from the Debye speed extracted from NRIXS (Nuclear Resonant Inelastic X-ray Scattering) measurements are underestimated compared to the material’s true seismic wave speeds. To illustrate the differences, the effects of various assumptions used to estimate the Debye speed, as well as seismic wave speeds, are examined with iron alloys at Earth’s inner core conditions. For the case of pure iron, the variation of the crystal orientation relative to the incoming X-ray beam causes a 40 % variation in the measured Debye speed, and leads to 3% and 31% underestimation in the compressional and shear wave speeds, respectively. Based upon various iron alloys, the error in the inferred seismic shear wave speed strongly depends upon the strength of anisotropy that can be quantified. We can also derive Debye speeds based upon seismological observations such as the PREM (Preliminary Reference Earth Model) and inner core anisotropy model. We show that these seismically derived Debye speeds are upper bounds for Debye speeds obtained from NRIXS experiments and that interpretation of the Debye speeds from the NRIXS measurements in terms of seismic wave speeds should be done with utmost caution.

Geophysical Inverse Theory and Regularization Problems

Inversion of pressure observations: an integral formulation

Recovering the isometry type of a Riemannian manifold from local boundary diffraction travel times

A new method for measuring surface acoustic wave speeds by acoustic microscopes and its application in characterizing laterally inhomogeneous materials

We define a new generalized inverse (named the gDMP inverse) for a Hilbert space operator using its generalized Drazin inverse and its Moore–Penrose inverse. Thus, we extend the DMP inverse for a square matrix to more general case. Also, we introduce two new classes of operators, g -EP and g -normal operators which include, respectively, EP operators and normal operators. A new binary relation is associated with the gDMP inverse is presented and studied. The notion of core-EP inverse for matrices is extended to generalized Drazin invertible operators on Hilbert space.